CORE Metadata, citation and similar papers at core.ac.uk

Provided by Kyoto University Research Information Repository

Title On the moduli of Kähler-Einstein Fano manifolds

Author(s) Odaka, Yuji

Citation 代数幾何学シンポジウム記録 (2013), 2013: 112-126

Issue Date 2013

URL http://hdl.handle.net/2433/214993

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University 代数幾何学シンポジウム記録 1 2013年度 pp.112-126

ON THE MODULI OF KAHLER-EINSTEIN¨ FANO MANIFOLDS

YUJI ODAKA

Abstract. We prove that K¨ahler-EinsteinFano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q- Fano varieties which should be put on the boundary of its canonical compactification.

1. Introduction After the well-known construction of moduli of stable curves, re- cently the construction of moduli of canonically polarized varieties (thus of general type) of higher dimension is established by the works of Shepherd-Barron, Koll´ar,Viehweg, Alexeev and subsequent authors, which depends on birational geometric techniques after the , admitting semi-log-canonical singularities. On the other hand, there are no general theory of constructing mod- uli of anti-canonically polarized varieties (i.e. of Fano type) nor of more general polarized varieties. In this exposition, we discuss a canonical way of constructing moduli using some differential geometry. That is, we will show the proof of the following theorem. Theorem 1.1. K¨ahler-EinsteinFano n-folds with finite automorphism groups form Hausdorff moduli algebraic space in the sense of Artin [Art71], which is actually a (complex) orbifold. As originally expected, the existence of K¨ahler-Einsteinmetrics on a Fano manifold is proved to be equivalent to the K-stability. The proof of stability of KE Fano manifolds is established by [Mab08], [Mab09], [Ber12] and for the converse direction, please consult [CDS12a], [CDS12b], [CDS13], [Tia12b] which discuss after the Donaldson’s pro- posal of a new continuity method using conical singular K¨ahler-Einstein metrics [Don11]. Thanks to the equivalence, we can paraphrase as the following. Theorem 1.2 (Re-statement). Smooth K-stable Fano varieties form separated moduli algebraic space and moreover we have only quotient singularities on it.

-112- 2

At the first talk at Kinosaki symposium (2010), the author sug- gested more general conjecture (K-moduli) - to construct moduli of K-(poly)stable varieties - which we review and discuss at the final sec- tion. So in a sense, this is a progress report on the conjecture. This principle of constructing moduli of these varieties with canoni- cal metrics has been often hided but existed behind the whole history of moduli construction in a sense. To see a bit of the connection, note that K¨ahler-Einsteinmetrics exist on smooth projective curves, smooth canonical models, and Calabi-Yau manifolds due to the Aubin- Yau existence theory ([Aub78], [Yau78]) while we know the existence of quasi-projective moduli of them by algebro-geometric independent work [Vie95]. What we discuss here is how to directly use K¨ahler- Einstein geometry for the construction of moduli as varieties or alge- braic spaces. Personally speaking, the author’s original motivation for this pro- gram of constructing moduli is from his algebro-geometric observa- tions on the K-stability [Od08], [Od11], which connected geometrically compactified moduli variety of general type varieties of the Koll´ar- Shepherd-Barron-Alexeev type (cf. e.g. [Kol10]) and the K-stability [Don02]. As originally expected, the existence of K¨ahler-Einstein metrics on a Fano manifold is proved to be equivalent to the K- stability. The proof of stability of KE Fano manifolds is established by [Mab08], [Mab09], [Ber12] and for the converse direction, please consult [CDS12a], [CDS12b], [CDS13], [Tia12b] which discuss after the Donaldson’s proposal of a new continuity method using conical singular K¨ahler-Einsteinmetrics [Don11]. While the equivalence is now established, the K-polystability (or the existence of K¨ahler-Einsteinmetric) for a given or a concrete Fano man- ifold, and the behaviour with respect to complex deformation have both certain independent aspects and they are difficult interesting questions in general which remain. To that question, we can prove the following. Theorem 1.3. For a flat projective family of Fano manifolds π : X → S, then the subset:

{s ∈ S | Xs is a KE Fano manifold with # Aut(X) < ∞} ⊂ S is a Zariski open subset of S. Note that this Zariski openness question is also discussed in Donald- son’s informal document [Don09] which takes a different approach un- der several conjectural hypotheses. Indeed, some discussion with Don- aldson on this question is also reflected in this notes as we will explain

-113- 3

at each point. A fundamental remark is that this Zariski openness does not hold without the condition of discreteness of automorphism group. Coming back to the moduli construction, the remained problem af- ter Theorem 1.1, 1.2 is to construct the compactification of them by attaching Gromov-Hausdorff limits. Indeed the following result estab- lished those algebraicity. Theorem 1.4. Gromov-Hausdorff limit of K¨ahler-EinsteinFano n- folds is a KE Q-Fano variety, i.e. log-terminal Fano varieties with a singular K¨ahler-Einsteinmetric. Note that the meaning of Gromov-Hausdorff convergence is enhanced there to encode the continuity of complex structures. See [DS12] for the details. In the case of Del Pezzo surfaces, the above algebraicity was estab- lished by Tian two decades ago [Tia90]. In that case, we succeeded in the construction of the compactification in explicit way in the joint work with Cristiano Spotti and Song Sun [OSS12]. This is pioneered by Mabuchi-Mukai [MM93] where degree 4 case is written. 2. Proofs Proof of Theorem 1.3. To state the proof, we need to recall the fol- lowing characterization of test configurations and introduce a certain modification of the K-stability notion. Proposition 2.1 ([RT07, Proposition 3.7]). Let X be a Fano mani- fold. Then for a large enough m ∈ Z>0, a one-parameter subgroup of 0 GL(H (X, OX (−mKX ))) is equivalent to the data of a test configura- 1 tion (X , L) of (X, −KX ) whose polarization L is very ample (over A ) with exponent m.

Definition 2.2. A Q-Fano variety X is said to be Km-stable (resp., Km-semistable) if and only if DF(X , L) > 0 for any test configuration of the above type (i.e., with very ample L of exponent m) which is non-trivial test configuration whose central fiber is normal. Notice that what [LX11] proved can be rephrased as that a Q-Fano variety is K-stable (resp., K-polystable, K-semistable) if and only if it is Km-stable (resp., Km-polystable, Km-semistable) for sufficiently divisible m ∈ Z>0. The name “Km-stability” followed a suggestion of Professor Simon Donaldson in a discussion. The equivalent notion was also treated in [Tia12a]. Let us also recall the definition of the algebro-geometric counter- part of the α-invariant ([Tia87]) i.e., the so-called global log-canonical threshold ([Sho92]) and its variants as follows.

-114- 4

Definition 2.3. Suppose X is a projective manifold, E is an effective Q-divisor on X, and L is an on X. Then we consider the following invariants.

lct(X,E) := sup{α > 0 | (X, αE) is log canonical},

( 1 ) lct (X; L) := inf ∈| |lct X, D for l ∈ Z , l D lL l >0 and

glct(X; L) := infl∈Z>0 lctl(X; L), which are called the log canonical threshold of X with respect to E, the log canonical threshold of X with respect to L with exponent l, and the global log canonical threshold of X with respect to L. For a Fano manifold X, we simply write glct(X) := glct(X; −KX ), lctl(X) := lctl(X; −KX ). According to the approximation theory of K¨ahler metrics by Bergman metrics, the last notion corresponds to α-invariant (cf. [Tia87], [CSD08, Appendix]). We use the following general proposition in what follows. Proposition 2.4. Given a flat family of Fano manifolds X → S and its relative pluri-anticanonical divisors D ∈ | − λKX /S|, we have uni- X − D − | form positive lower bounds for both lct( s; KXs ) and lct( s; KXs Ds ) where s runs through S. proof of Proposition 2.4. First, let us recall the following basic lemma. Lemma 2.5. For any log pair (X,D) with Q-Cartier effective Q- divisor D, ≥ lct(X,D) 1∑/mult p(D), Q where if D is an effective∑ -divisor aiDi with prime divisors Di and ai ∈ Q>0, mult p(D) := aimult p(Di). This Lemma follows from the upper semicontinuity of the log canonical thresholds (cf. e.g., [Mus02]), applied to a degenerating family of D to a multiple of a smooth local divisor (cf. e.g., [Kol97] for more details). On the other hand, supposing −λKX is very ample for every Fano n-fold X, we have n n mult p(D) ≤ λ (−KX ) for all p ∈ D with D ∈ | − λKX |. Indeed, if we take general members D1, ··· ,Dn−1 among | − λKX | which pass through the point p ∈ X without singularity at p, the inequality follows from the standard esti- n n mate of the intersection number (D1. . . . .Dn−1.D) = λ (−KX ) . The proof of Proposition 2.4 is completed. 

-115- 5

On the other hand, let us recall the following results from [OS11]. Theorem 2.6 ([OS11, Theorem 4.1]). Let X be a , D′ is an effective Q divisor and 0 ≤ β ≤ 1. (i) Assume (X, (1−β)D) is a log Calabi-Yau pair, i.e., KX + (1 − β)D is numerically equivalent to the zero divisor and it is a semi-log-canonical pair (resp. kawamata-log- terminal pair). Then, ((X,D),L) is logarithmically K-semistable (resp. logarithmically K-stable) with respect to the boundary’s parameter β. (ii) Assume (X, (1 − β)D′) is a semi-log-canonical model, i.e., ′ KX + (1 − β)D is ample and it is a semi-log-canonical pair. Then, ′ ((X,D),KX + (1 − β)D ) and β ∈ Q>0 is log K-stable with respect to the boundary’s parameter β. Theorem 2.7 (cf. [OS11, Corollary 5.5]). Let X be an ar- bitrary Q-Fano variety and D is an integral pluri-anti-canonical Cartier divisor D ∈ | − λKX | with some λ ∈ Z>0. Then (X, −KX ) is logarithmically K-stable (resp. logarithmically( K- λ−1 − semistable) for cone angle 2πβ with )λ < β < λ (1 + n+1 { − − | } λ−1 ≤ ≤ − ( n )min glct(X; KX ), glct(D; KX D) /λ )(resp. λ β λ n+1 { − − | } 1 + ( n )min glct(X; KX ), glct(D; KX D) /λ). We make some remarks. Recall that Theorem 2.6 (i) extends and algebraically recovers [Sun11, Theorem 1.1], [Bre11, Theorem1.1] and [JMR11, Theorem 2]. In other words, at least for the case where X is smooth, D is a smooth integral divisor, the above is expected to follow from their results as well. Regarding Theorem 2.7, the original version of [OS11, Corollary 5.5] treated only λ = 1 case, but it is recently noticed that it can be simply extended to the above form. We also remark that the case if X and D are both smooth and λ = 1, Theorem 2.7 also follows from [Ber10, Theorem 1.8] combined with [Ber12, Theorem1.1]. The above two theorems imply that Corollary 2.8. For an arbitrary smooth Fano manifold X and a smooth pluri-anticanonical divisor D ∈ |−λKX |, (X, −KX ) is logarith- mically K-stable (resp. logarithmically K-semistable) for cone angle 2πβ − n+1 { − − | } with 0 < β < (λ 1 + ( n )min glct(X; KX ), glct(D; KX D) )/λ ≤ ≤ − n+1 { − − | } (resp. 0 β (λ 1+( n )min glct(X; KX ), glct(D; KX D) )/λ). Combining the above lower boundedness of log canonical thresholds and Corollary 2.8, we can take uniform β∞(> 0) such that every Fano manifolds Xs (s ∈ S) are log K-stable for cone angle 2πβ∞ with respect to the divisor Cartier Ds. We use the following theorem for the proof as well.

-116- 6

Theorem 2.9. [CDS12b, Theorem 1], [CDS13, Theorem 2, the para- graph below it] For a Fano manifold X, if X does not admit K¨ahler- Einstein metric, the following holds. For a smooth divisor D ∈ |−λKX | with some λ ∈ Z>0, we have a non-trivial normal log test configuration X D − − 1 ≤ ( , ) of (X, (1 β∞)D) with cone angle 2πβ∞ (1 λ < β∞ 1) such X D − X Q that DFβ∞ (( , ); KX ) = 0 and 0 is a -Fano variety. Moreover, the Q-Gorenstein index of X0 can be taken uniformly if we have a fixed dimension dim(X), λ and a lower bound for β∞ which is bigger than − 1 1 λ .

Note that the bound of index of X0 is equivalent to that they are all parametrized in a fixed , by the recent boundedness result [HMX12, Corollary 1.8]. Note that the analogical bound of Q- Gorenstein index of the limit as Theorem 2.4 was indeed already proved earlier in non-log setting as well ([DS12]). Thanks to the existence of the uniform β∞, in turn Theorem 2.9 implies that there is a large enough but fixed m such that for any s ∈ S, Xs is K¨ahler-EinsteinFano manifold if and only if it is Km- stable. Take a Hilbert scheme H which exhausts all Xs, embedded by | − | mKXs . Automatically we have CM line bundle on H which we 0 − denote by ΛCM and there are actions of SL(h ( mKXs )) on both H and ΛCM (linearization). Denote the locus in H which parametrizes K¨ahler-EinsteinFano manifold Xs with s ∈ S by HKE, and the locus which parametrizes normal fibers by Hnormal. It is well known that Hnormal is a Zariski open subset of H as a general phenomenon. 0 − Let us fix a maximal torus T of SL(h ( mKXs )). Recall the following known fact for usual setting in Geometric Invariant Theory [Mum65] which is for ample linearized line bundles. Lemma 2.10 (cf. [Mum65, Chapter 2, Proposition 2.14]). Consider the action of a reductive algebraic group G on a polarized projective scheme (H, Λ) i.e. H is a projective scheme and Λ is an ample lin- earized line bundle. Let us denote the corresponding GIT weights func- tion with respect to a one parameter subgroup φ: Gm → G at h ∈ H by λ(h, φ; Λ). Then there are some finite linear functions {li}i∈I on Homalg.grp(Gm,T )⊗Z R with rational coefficients, indexed by a finite set I such that the weight function λ(h, −; Λ), regarded as a function from Homalg.grp(Gm,T ) to Z, extends to a piecewise linear rational function of the form sup{lj(−) | j ∈ Jh} I with some Jh ⊂ I. Moreover, ψ : H → 2 which maps h to Jh, is constructible in the sense that for any J ∈ I, {h ∈ H | Jh = J} is constructible.

-117- 7

The last line is not stated in [Mum65, Proposition 2.14] but follows rather easily from the proof and the arguments written there. Recall that the CM line bundle [PT06] (cf. also [FS90]) is a line bun- dle defined on the base scheme for an arbitrary polarized family and the formal GIT weight of the CM line bundle on the Hilbert scheme is exactly the corresponding Donaldson-Futaki invariant. However, un- fortunately we can not directly apply to the CM line bundle ΛCM which is not necessarily ample in general. Nevertheless, Lemma 2.10 can be still extended to the case for our H and the CM line bundle ΛCM on it, though λ(t, −;ΛCM ) is not necessarily convex. To see that, first we take a Pl¨ucker polarization ΛP l on H which is known to be very ample, yielding the Pl¨ucker embedding of Hilbert scheme. Take sufficiently ∈ Z ⊗ ⊗l large l >0 such that ΛCM ΛP l is also very ample. Then the point − − ⊗ ⊗l − − is that λ(h; ;ΛCM ) = λ(h; ;ΛCM ΛP l) lµ(h; ;ΛP l). Therefore Lemma 2.10 extends to the case with our setting that Hilbert scheme 0 − is H, L = ΛCM and G = SL(h ( mKXs )) as follows.

Lemma 2.11. There are some finite linear rational functions {li}i∈I′ on Homalg.grp(Gm,T ) ⊗Z R with rational coefficients, indexed by a fi- nite set I′ such that the followings hold. For each [X ⊂ P], there ′ is JX ⊂ I such that its Donaldson-Futaki invariants with respect to 0 − one parameter subgroups of G := SL(h ( mKXs ) regarded as a func- tion from Homalg.grp(Gm,T ) to Z, extends to a piecewise linear ratio- { } ⊂ ′ nal function whose pieces are lj j∈JX with some JX I . More- ′ over, ψ is constructible in the sense that for any subset JX ⊂ I , {[X ⊂ P] ∈ H | JX = J} is constructible. Next, let us discuss from another viewpoint which is more birational- geometric. This is related to what Donaldson called “splitting of orbits” in [Don09]. Considering the family over S and trivialize the locally free π∗O(D) on a Zariski open covering {Si}i we have the corresponding morphism (⊔Si × T ) → H, which can be regarded as 1 r a rational map (⊔Si × (P ) ) 99K H. For simplicity, re-set S := ⊔Si from now on which does not lose generality from our assertion of the theorem. Thinking of a resolution of indeterminacy, we get some blow up of S × (P1)r along a closed subscheme inside S × ((P1)r \ T ). The blow up is generically isomorphic over S × ((P1)r \ T ), which means the GIT weights with respect to any fixed one parameter subgroups in T stay constant for some Zariski open dense subset S′ of S. Replace S by S \ S′ then we can construct a constructible stratification of S such that each stratum’s degenerations with respect to any fixed one parameter subgroup of G fits into a flat family again. This implies that those corresponding Donaldson-Futaki invariants are constant on each

-118- 8

strata by [PT06]. We can also prove it by using Wang’s interpretation of Donaldson-Futaki invariant [Wan12]. Claim 2.12. There is a finite set I of one parameter subgroups {φi}: Gm → T (i ∈ I) such that the following is true. There is a stratification of Hnormal by constructible subsets {HJ }J⊂I , such that for any t ∈ HJ ⊂ Hnormal, Xt is Km-stable (resp., Km-semistable) if and only if DF(φj;[Xt ⊂ P]) > 0 (resp., DF(φj;[Xt ⊂ P]) ≥ 0) for all j ∈ J where DF(φj;[Xt ⊂ P]) actually only depends on j and J. Here DF(φj;[Xt ⊂ P]) means the Donaldson-Futaki invariant of the test con- X O O − figuration of ( t, (1) = ( mKXt )) associated to the one parameter subgroup via Proposition 2.1. Summarizing up, we proved that the locus of Fano manifolds which are Km-stable (resp., Km-semistable, Km-unstable, not Km-stable) with respect to the maximal torus T ’s action are all constructible subsets. Then the constructibility of Km-stable locus in H follows from C. Chevalley’s lemma (cf. [Har77]) which states that the im- age of constructible set by an algebro-geometric morphism is again constructible. Indeed, we can apply it to the group action morphism 0 − × → SL(h ( mKXs )) H H and the constructible subset of H which parametrizes Fano manifolds which are not Km-stable with respect to the action of T . Then, it shows that the subset of H which parametrizes Fano manifolds with discrete automophism groups which are not Km- stable is again constructible. On the other hand, using the deformation theory of K¨ahlermetrics with constant scalar curvature it was shown in [LS94] that

{s ∈ S | Xs is a KE Fano manifold with # Aut(X) < ∞} ⊂ S is an open subset with respect to the Euclidean topology. More pre- cisely speaking, we need to take resolution of singularities of S to apply [LS94]. Thus the locus of K¨ahler-EinsteinFano manifolds with finite automorphism groups should be a Zariski open subset. We complete the proof of the main theorem.  We prove Theorem 1.1 using Theorem 1.3 as follows. proof of Theorem 1.1 (using Theorem 1.3). Consider the Hilbert scheme H which exhausts all m-pluri-anticanonically embedded Fano n-folds with uniform m ∈ Z>0. By applying the previous statement of Theorem 1.3 to a Zariski open covering of universal family over H, we obtain that the locus which parametrizes K¨ahler-EinsteinFano manifolds form a Zariski open locus. Therefore the natural moduli

-119- 9

op functor F : (Sch/C) → (Sets) whose B-valued points set is defined by { Y → | Y ⊂ P 0 − ∈ ∀ ∈ } ∼ flat proj. f : B [ b (H ( mKYb ))] HKE for b B / = naturally gives rise to Deligne-Mumford stack which is simply a quo- 0 − tient stack [HKE/ PGL(h ( mKXs ))]. Then the Keel-Mori theorem [KM97, Corollary 1.3] implies that we have a coarse moduli algebraic space for it. The separatedness follows from a standard argument using the uniqueness of the limits as Riemannian manifolds (cf. e.g. [CS10, Lemma 8.1]) and the Bando-Mabuchi theorem [BM85], or [Don01]. Moreover, the moduli algebraic stack (in the sense of Deligne- Mumford) is smooth as the deformation is unobstructed. It is be- 2 2 n−1 cause of the vanishing of obstruction group H (X,TX ) = H (X, Ω ⊗ −1 KX ) = 0 thanks to the Akizuki-Nakano’s generalization of the Ko- daira vanishing [AN54] combined with the finiteness of isotropy groups. Therefore the coarse moduli algebraic space has only quotient singu- larities. This completes the proof of Theorem 1.1. 

3. Revisiting the general conjecture The moduli space constructed in Theorem 1.1 is expected to have canonical projective compactification by attaching (singular) Q-Fano varieties on the boundary (cf. [OSS12]) but let us review this in broader context as in [Od10]. The author expects that K-semistable polarized varieties have coarse projective moduli in general. Including what we have already mentioned, the inspirations are mainly from the following backgrounds. • The moment map picture on the infinite-dimensional space of complex structures by Fujiki, Donaldson ([Fuj90], [Don97], [Don04]). • The observations ([Od08], [Od11]) that the projective moduli of general type varieties with semi-log-canonical singularities, constructed by using the minimal model program (cf. [Kol10]) and some other projective moduli corresponds to K-stability. • The algebraicity of Gromov-Hausdorff limits of K¨ahler- Einstein Fano manifolds ([DS12]) and the Gromov (pre)compactness theorem (cf. [MM93], [Spo12], [OSS12]). Or we can also say that the Yau-Tian-Donaldson conjecture itself is another background, as GIT polystably is originally designed by Mumford [Mum65] to make them form projective moduli although the K-stability is not quite a GIT stability notion in that original sense.

-120- 10

As a preparation for making the expectation rigorous, we introduce the following terminology. Let us call a numerical equivalence class of ample line bundle, a weak polarization and a pair of projective variety and weak polarization a weakly polarized variety. Then the following is a modified version of the conjecture stated in the Kinosaki symposium 2010. Conjecture 3.1 (K-moduli cf. [Od10]). The moduli functor of K- semistable weakly polarized varieties has proper coarse moduli algebraic 1 ¯ space McscK . It set-theoritically parametrizes the set of all smooth K-polystable weakly polarized varieties and certain equivalence classes of singular weakly polarized varieties with singular cscK metrics. The ¯ connected components of McscK are all projective schemes, because the CM line bundle descends to ample line bundles on them. We refer to [OSS12] for detailed discussions on our Q-Fano varieties case which largely settled (smoothable) del Pezzo surfaces case explic- itly. In that paper, we have constructed explicit moduli for all degrees as the Gromov-Hausdorff compactifications. Indeed, the author ex- pects more generally the K-moduli can be constructed as a “Gromov- Hausdorff compactification” in appropriately rescaled sense, of moduli of smooth cscK polarised varieties. The point of view with Gromov-Hausdorff limit had been missed in the original K-moduli expectation [Od10] and inspired by differential geometric colleagues later (e.g. [DS12], [Spo12]), which the author is truly greatful. Especially the memorable collaboration with Spotti and Sun [OSS12] makes the author more confident on the general conjec- ture above. Moreover, very recently the wonderful work of Berman- Guenancia [BG13] -extending hyperbolic metrics on stable curves- ap- peared which constructed singular K¨ahler-Einstein(thus cscK) metrics on semi-log-canonical varieties with ample canonical classes. We note that allowing just the semi-log-canonical singularities should be more general phenomenon for all classes of K-semistable (weakly) polarized varieties [Od08].

4. About the limits An interesting problem is to recover or to have better understand- ing of the Gromov-Hausdorff limits of K-stable punctured family or maximally destabilizing test configurations of K-unstable varieties via the minimal model program along the line of [Od08], [LX11]. (The

1With infinitely many connected components.

-121- 11

author expects that the destabilizing test configuration of non-slc vari- eties [Od08] via MMP is maximal when we normalize Donaldson-Futaki invariants. ) The following characterization for special case is essentially obtained in [OS12], which the author was not aware for a while. Theorem 4.1 (CM minimizing for (special) Fano varieties). Sup- pose π : X → C ∋ 0 is a Q-Gorenstein flat projective family of n- dimensional Q-Fano varieties over a projective curve C and the alpha X − invariant of the central fiber glct( 0 ; KX0 ) is at least (resp. bigger n than) n+1 . Regarding π as a polarized family but putting anticanonical line bundle on X , any other flat polarized family (X ′, L′) → C which is isomorphic to X → C away from the central fiber, we have ′ ′ deg(CM(X , −KX /C)) ≤ (resp. <)deg(CM(X , L )). In general, the author expect that the - a priori differential geometric notion - the Gromov-Hausdorff limit of punctured K-stable family of Q-Fano varieties is, algebraically, characterized as what minimizes the CM degree as above, and the construction or above characterization is to be able to prove via the minimal model program or differential geometrically K¨ahler-Ricciflow. An easier version may be trivial family of K-semistable Fano varieties case (which, at least in smooth case by [CDS13], [Tia12b], isotrivially degenerates to K-polystable Fano manifolds). For the proof of Theorem 4.1, first we take a resolution of indeter- ′ ⊗(−r) minancy of X 99K X as blow up π : B → X where KX (−E) with the Cartier exceptional divisor E is the pullback of L′⊗(−r). Then we can estimate ′ ′ ⊗(−r) ∗ ⊗(−r) CM (X , L ) − CM (X ,KX ) = CM(B, π KX (−E)) − CM(X , L), the difference of CM lines of X and X ′, in the completely similar way 1 as in [OS12] in which they did estimate for the case X is X0 × P via the basics of discrepancy. Analogously, for Calabi-Yau varieties and general type varieties, if we apply the estimates of Donaldson-Futaki invariants at [Od11] similarly to the differences of CM degrees of a family and its blow up, which we can obtain as a resolution of rational map X 99K X ′, we can (re)prove the following. The semi-log-canonical models (“general type”) case was obtained by X. Wang - C. Xu earlier [WX12], so in a sense, our results are its generalization to different types of varieties. Theorem 4.2 (Analogues). (i) (Calabi-Yau case) Suppose π : X → C ∋ 0 is a Q-Gorenstein flat projective family of

-122- 12

n-dimensional semi-log-canonical models over a projective curve C. Regarding π as a polarized family but putting canonical line bundle on X , any other flat polarized family (X ′, L′) → C which is isomorphic to X → C away from the central fiber, we have ′ ′ deg(CM(X , −KX /C)) ≤ (resp. <)deg(CM(X , L )). (ii) (Canonical model case: [WX12, Theorem 6]) Suppose π : X → C ∋ 0 is a Q-Gorenstein flat projective (“KSBA”) family of n- dimensional semi-log-canonical models over a projective curve C. Regarding π as a polarized family but putting canonical line bundle on X , any other flat polarized family (X ′, L′) → C which is isomorphic to X → C away from the central fiber, we have ′ ′ deg(CM(X , −KX /C)) ≤ (resp. <)deg(CM(X , L )). Corollary 4.3 (well known? e.g. [Bou13]). If a punctured family of polarized log terminal Calabi-Yau varieties can be completed with a log terminal CY filling, there is no other semi-log-canonical CY filling. In particular, log terminal polarized Calabi-Yau varieties have Haus- dorff moduli. We end this short article by saying regretfully that the above CM minimizing result was once claimed with Richard Thomas for any K- (semi)stable limit (family) whose proof had a gap. We still hope a possibility to recover that in general situation. Acknowledgments. The author appreciates the organizers of the Ki- nosaki symposium 2013, S. Kuroda, K. Yamada, K. Yoshioka, for invit- ing him to give the opportunity. The author would like to thank Professor S. Donaldson for useful dis- cussions and warm encouragements. He also would like to thank Pro- fessors I. Cheltsov, C. Xu, V. Tosatti, S. Boucksom, Doctors S. Song, C. Spotti, for useful discussions and comments, Professor G. Tian for kindly sending the pdf file of [Tia12a].

References

[AN54] Y. Akizuki, S. Nakano, Note on Kodaira-Spencer’s proof of Lefschetz the- orems, Proc. Jap. Acad. vol. 30 (1954), 266―272. [Art71] M. Artin, Algebraic spaces, Yale Mathematical Monographs vol. 3 Yale University Press (1971). [Aub78] T. Aubin, Equations´ du type Monge-Amp´eresur les vari´et´esk¨ahl´eriennes compactes, Bull. Sci. Math. (2) vol. 102 (1978), no. 1, 63―95.

-123- 13

[BM85] S. Bando, T. Mabuchi, Uniqueness of Einstein K¨ahlermetrics modulo con- nected group actions, “, Sendai 1985”, Adv. Stud. Pure Math., vol. 10, Kinokuniya and North-Holland, (1987), 11-40. [Ber10] R. Berman, A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and K¨ahler-Einstienmetrics, arXiv:1011.3976. [Ber12] R. Berman, K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics, arXiv:1205.6214. [BG13] R. Berman, H. Guenancia, K¨ahler-Einsteinmetrics on stable varieties and log canonical pairs, arXiv:1304.2087. [Bou13] S. Boucksom, Private communication. [Bre11] S. Brendle, Ricci-flat K¨ahler metrics with edge singularities, arXiv:1103.5454. [CDS12a] X. Chen, S. Donaldson, S. Sun, K¨ahler-Einsteinmetrics on Fano mani- folds, I: approximation of metrics with cone singularities. arXiv:1211.4566. [CDS12b] X. Chen, S. Donaldson, S. Sun, K¨ahler-Einsteinmetrics on Fano mani- folds, II: limits with cone angle less than 2Π. arXiv:1212.4714. [CDS13] X. Chen, S. Donaldson, S. Sun, K¨ahler-Einsteinmetrics on Fano man- ifolds, III: limits as cone angle approaches 2 π and completion of the main proof. arXiv:1302.0282. [CSD08] I. Cheltsov and C. Shramov, Log canonical thresholds of smooth Fano threefolds. (with an appendix by J. P. Demailly), Uspekhi Mat. Nauk, vol. 63 (2008), 73–180. arXiv:0806.2107 (2008). [CS10] X. Chen, S. Sun, Calabi flow, geodesic rays, and uniqueness of constant scalar curvature K¨ahlermetrics, arXiv:1004.2012. [Don97] S. K. Donaldson, Remarks on gauge theory, and 4- manifold topology, Fields Medallists’Lectures (Atiyah, Iagolnitzer eds. ) World Scinetific (1997), 13–33. [Don01] S. K. Donaldson, Scalar curvature and projective embeddings I., J. Diff. Geom. vol. 59, (2001) 479–522. [Don02] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Dif- ferential Geom. vol. 62, (2002) 289–349. [Don04] S. K. Donaldson, Conjectures in K¨ahlergeometry, Strings and geometry, Clay Math. Proc. , vol. 3, Amer. Math. Soc. , Providence, RI, (2004), 71–78. [Don09] S. K. Donaldson, Discussion of the K¨ahler-Einsteinproblem, an informal document, available on his webpage. [Don11] S. K. Donaldson, K¨ahlermetrics with cone singularities along a divisor, arXiv:1102.1196. [DS12] S. Donaldson, S. Sun, Gromov-Hausdorff limits of K¨ahlermanifolds and algebraic geometry, arXiv:1206.2609. [Fuj90] A. Fujiki, The moduli spaces and K¨ahlermetrics of polarized algebraic varieties Sugaku vol. 42 (1990), 231-243. [FS90] A. Fujiki, Schumacher, The Moduli Space of Extremal Compact K¨ahler Manifolds and Generalized Weil-Petersson Metrics, Publ. R. I. M. S. Kyoto Univ. vol. 26 (1990), 101-183. [HMX12] C. Hacon, J. Mckernan, C. Xu, ACC for log canonical thresholds, arXiv:1208.4150. [Har77] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 82, Springer Verlag (1977).

-124- 14

[JMR11] T. Jeffres, R. Mazzeo, Y. Rubinstein, K¨ahler-Einsteinmetrics with edge singularities, arxiv: 1105.5216. [KM97] S. Keel, S. Mori, Quotients by groupoids, Ann. of Math. vol. 145, (1997) 193–213. [Kol97] J. Koll´ar,Singularities of pairs, Algebraic geometry―Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, (1997), 221–286. [Kol10] J. Koll´ar,Moduli of varieties of general type, arXiv:1008.0621. [LS94] C. LeBrun, S. R. Simanca, Extremal K¨ahlermetrics and complex deforma- tion theory, Geom. Funct. Anal. vol. 4 (1994) [LX11] C. Li, C. Xu, Special test configurations and K-stability of Fano varieties, to appear in Annals of Math. [Mab08] T. Mabuchi, K-stability of constant scalar curvature polarization, arXiv:0812.4903 (2008). [Mab09] T. Mabuchi, A stronger concept of K-stability, arXiv:0910.4617 (2009). [MM93] T. Mabuchi, S. Mukai, Stability and Einstein-K¨ahlermetric of a quartic del Pezzo surface, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, (1993), 133–160. [Mum65] D. Mumford, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, (1965). [Mus02] M. Mustata, Singularities of pairs via jet schemes, J. Amer. Math. Soc. vol. 15 (2002) [Od08] Y. Odaka, The GIT stability of polarized varieties via discrepancy, Annals of Math. vol. 177, iss. 2, 645-661. (2013). [Od10] Y. Odaka, On the GIT stability of polarized varieties -a survey-, Proceeding of Kinosaki Algebraic Geometry symposium (Oct, 2010). The revision available at https://sites.google.com/site/yujiodaka2013/index/proceeding [Od11] Y. Odaka, The Calabi conjecture and K-stability, Int. Math. Res. Notices vol. 13, (2011). [OS11] Y. Odaka, S. Sun, Testing log K-stability via blowing up formalism, arXiv:1112.1353. [OS12] Y. Odaka, Y. Sano, Alpha invariants and K-stability of Q-Fano varieties, Adv. in Math. vol. 229, (2012) [OSS12] Y. Odaka, C. Spotti, S. Sun, Compact moduli spaces of del Pezzo surfaces and K¨ahler-Einsteinmetrics, arXiv:1210.0858. (2012). [PT06] S. Paul, G. Tian, CM Stability and the Generalized Futaki Invariant I, arXiv:0605278. [RT07] J. Ross, R. Thomas: A study of the Hilbert Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), 201-255. [Sho92] V. Shokurov, Three-dimensional perestroikas, Izvestiya: Mathematics vol. 56 (1992). [Spo12] C. Spotti, Degenerations of K¨ahler-EinsteinFano Manifolds (thesis), Im- perial College London (2012). arXiv:1211.5334. [Sun11] S. Sun, Note on K-stability of pairs, to appear in Math. Ann. arXiv:1108.4603. [Tia87] G. Tian, On K¨ahler-Einsteinmetrics on certain K¨ahlermanifolds with C1(M) > 0, Invent. Math. vol. 89, (1987) 225–246.

-125- 15

[Tia90] G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. vol. 101 (1990), no. 1, 101-172. [Tia97] G. Tian, K¨ahler-Einsteinmetrics with positive scalar curvature, Invent. Math. vol. 130, (1997) 1–37. [Tia12a] G. Tian, Existence of Einstein metrics on Fano manifolds, Progress in Mathematics, vol. 297 (2012). [Tia12b] G. Tian, K-stability and K¨ahler-Einsteinmetrics, arXiv:1211.4669. [Vie95] E. Viehweg, Quasi-projective Moduli for Polarized Manifolds, Springer- Verlag, (1995). [Yau78] S. T. Yau, On the Ricci curvature of a compact K¨ahlermanifold and the complex Monge-Amp`ereequation. I, Comm. on Pure and Applied Math. vol. 31 no. 3, (1978), 339–411. [Wan12] X. Wang, Heights and GIT weights, Math. Res. Letters vol. 19, (2012). [WX12] X. Wang, C. Xu, Non existence of asymptotic GIT compactification, arXiv:1212.0173.

Department of Mathematics, Kyoto university, Oiwake-cho, Ki- tashirakawa, Sakyo-ku, Kyoto city, Kyoto, 606-8285, JAPAN E-mail address: [email protected]

-126-